Preheat oven to 400°F. Line 2 baking sheets with parchment paper.
In a bowl, whisk together the flour, cocoa powder, cream of tartar, baking soda, and salt. Set aside.
In a large mixing bowl or the bowl of a stand mixer, beat the butter until smooth and creamy, about 1 minute. Add the sugar and beat until light and fluffy, 2-3 minutes, scraping down the bowl as needed.
Beat in the eggs and vanilla until well combined, scraping down the bowl once or twice.
With the mixer on low, gradually add the dry ingredients and mix just until a stiff dough forms. Do not overmix.
In a shallow bowl, whisk together the sugar and cinnamon.
Use a small cookie scoop (about 1 1/2 tablespoons) to divide the dough into 24 equal portions.
Roll each portion into a ball, then roll in the cinnamon sugar to coat. Arrange the coated balls on the prepared baking sheets, spacing them about 2 inches apart.
Gently press each ball with the bottom of a drinking glass to flatten slightly to about 1/2-inch thickness.
Bake for 9 to 12 minutes, until the cookies are puffed, just set at the edges, and the tops look dry and slightly cracked. The centers will still look a bit soft.
Remove from oven and let the cookies cool on the baking sheet for 5 minutes.
Transfer to a wire rack to cool completely, about 20-30 minutes.